#369999

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2482
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1034 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,4,3)(5,6,10,15,11,16,9,14)(7,12,20,28,18,26,19,27)(8,13,23,31,22,30,24,32)(17,33,21,36,34,25,35,29);(1,3,4,2)(5,7)(6,12)(8,21,22,35)(9,19)(10,20)(11,18)(13,29,30,36)(14,27)(15,28)(16,26)(17,23,34,24)(25,31,33,32);(5,8,9,24,11,22,10,23)(6,13,14,32,16,30,15,31)(7,17,19,35,18,34,20,21)(12,25,27,36,26,33,28,29);(5,7)(6,12)(8,17)(9,19)(10,20)(11,18)(13,25)(14,27)(15,28)(16,26)(21,23)(22,34)(24,35)(29,31)(30,33)(32,36)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=eg,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^b=fi,e^c=ei,g^c=gi,f^d=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 277195);